3.201 \(\int \sqrt{c e+d e x} (a+b \cosh ^{-1}(c+d x)) \, dx\)

Optimal. Leaf size=127 \[ -\frac{4 b \sqrt{e} \sqrt{-c-d x+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right ),-1\right )}{9 d \sqrt{c+d x-1}}+\frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1} \sqrt{e (c+d x)}}{9 d} \]

[Out]

(-4*b*Sqrt[-1 + c + d*x]*Sqrt[e*(c + d*x)]*Sqrt[1 + c + d*x])/(9*d) + (2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c
+ d*x]))/(3*d*e) - (4*b*Sqrt[e]*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(9*d*Sqrt[
-1 + c + d*x])

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Rubi [A]  time = 0.0971451, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5866, 5662, 102, 12, 117, 116} \[ \frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1} \sqrt{e (c+d x)}}{9 d}-\frac{4 b \sqrt{e} \sqrt{-c-d x+1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )\right |-1\right )}{9 d \sqrt{c+d x-1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*e + d*e*x]*(a + b*ArcCosh[c + d*x]),x]

[Out]

(-4*b*Sqrt[-1 + c + d*x]*Sqrt[e*(c + d*x)]*Sqrt[1 + c + d*x])/(9*d) + (2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c
+ d*x]))/(3*d*e) - (4*b*Sqrt[e]*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(9*d*Sqrt[
-1 + c + d*x])

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rubi steps

\begin{align*} \int \sqrt{c e+d e x} \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{e x} \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(e x)^{3/2}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{3 d e}\\ &=-\frac{4 b \sqrt{-1+c+d x} \sqrt{e (c+d x)} \sqrt{1+c+d x}}{9 d}+\frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^2}{2 \sqrt{-1+x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{9 d e}\\ &=-\frac{4 b \sqrt{-1+c+d x} \sqrt{e (c+d x)} \sqrt{1+c+d x}}{9 d}+\frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac{(2 b e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{9 d}\\ &=-\frac{4 b \sqrt{-1+c+d x} \sqrt{e (c+d x)} \sqrt{1+c+d x}}{9 d}+\frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac{\left (2 b e \sqrt{1-c-d x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{9 d \sqrt{-1+c+d x}}\\ &=-\frac{4 b \sqrt{-1+c+d x} \sqrt{e (c+d x)} \sqrt{1+c+d x}}{9 d}+\frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac{4 b \sqrt{e} \sqrt{1-c-d x} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )\right |-1\right )}{9 d \sqrt{-1+c+d x}}\\ \end{align*}

Mathematica [C]  time = 0.43508, size = 131, normalized size = 1.03 \[ \frac{\sqrt{e (c+d x)} \left (\frac{2}{3} (c+d x)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )-\frac{4 b \left (\sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},(c+d x)^2\right )+c^2+2 c d x+d^2 x^2-1\right )}{9 \sqrt{\frac{c+d x-1}{c+d x}} \sqrt{c+d x+1}}\right )}{d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcCosh[c + d*x]),x]

[Out]

(Sqrt[e*(c + d*x)]*((2*(c + d*x)^(3/2)*(a + b*ArcCosh[c + d*x]))/3 - (4*b*(-1 + c^2 + 2*c*d*x + d^2*x^2 + Sqrt
[1 - (c + d*x)^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(9*Sqrt[(-1 + c + d*x)/(c + d*x)]*Sqrt[1 + c
+ d*x])))/(d*Sqrt[c + d*x])

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Maple [A]  time = 0.016, size = 194, normalized size = 1.5 \begin{align*} 2\,{\frac{1}{de} \left ( 1/3\, \left ( dex+ce \right ) ^{3/2}a+b \left ( 1/3\, \left ( dex+ce \right ) ^{3/2}{\rm arccosh} \left ({\frac{dex+ce}{e}}\right )-2/9\,{\frac{1}{e} \left ( \sqrt{-{e}^{-1}} \left ( dex+ce \right ) ^{5/2}+\sqrt{{\frac{dex+ce+e}{e}}}\sqrt{-{\frac{dex+ce-e}{e}}}{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ){e}^{2}-\sqrt{-{e}^{-1}}\sqrt{dex+ce}{e}^{2} \right ){\frac{1}{\sqrt{-{e}^{-1}}}}{\frac{1}{\sqrt{{\frac{dex+ce+e}{e}}}}}{\frac{1}{\sqrt{{\frac{dex+ce-e}{e}}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))*(d*e*x+c*e)^(1/2),x)

[Out]

2/d/e*(1/3*(d*e*x+c*e)^(3/2)*a+b*(1/3*(d*e*x+c*e)^(3/2)*arccosh(1/e*(d*e*x+c*e))-2/9/e*((-1/e)^(1/2)*(d*e*x+c*
e)^(5/2)+((d*e*x+c*e+e)/e)^(1/2)*(-(d*e*x+c*e-e)/e)^(1/2)*EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)*e^2-(-1/
e)^(1/2)*(d*e*x+c*e)^(1/2)*e^2)/(-1/e)^(1/2)/((d*e*x+c*e+e)/e)^(1/2)/((d*e*x+c*e-e)/e)^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))*(d*e*x+c*e)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{d e x + c e}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))*(d*e*x+c*e)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*e*x + c*e)*(b*arccosh(d*x + c) + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e \left (c + d x\right )} \left (a + b \operatorname{acosh}{\left (c + d x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))*(d*e*x+c*e)**(1/2),x)

[Out]

Integral(sqrt(e*(c + d*x))*(a + b*acosh(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d e x + c e}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))*(d*e*x+c*e)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*e*x + c*e)*(b*arccosh(d*x + c) + a), x)